Answer
410.1k+ views
Hint: To solve this question using the formula of capacitance between two parallel plates i.e. $ c = \dfrac{{A{\varepsilon _0}}}{d} $ and when any dielectric is inserted between the capacitor plates, the capacitance increases by a factor i.e. new capacitance $ {c_d} = \dfrac{{kA{\varepsilon _0}}}{d} $ . Now, compare both the equations and use the formula of permittivity that is $ \varepsilon = {\varepsilon _0}k $ and you will get the desired result.
Complete step by step solution
Let us take the original capacitance to be $ \prime c\prime $ and the new capacitance, when the dielectric is inserted between the plates, be $ {c_d}^{} $ .
Using the formula of capacitance, $ c = \dfrac{{A{\varepsilon _0}}}{d} $ ....... $ \left( 1 \right) $
When a dielectric is inserted between plates, the capacitance of capacitor increases by the factor of $ k $ (dielectric constant) i.e. the new formula is $ {c_d} = \dfrac{{kA{\varepsilon _0}}}{d} $ ............ $ \left( 2 \right) $
where $ A $ is the area of the capacitor is plates and $ d $ is the distance between the two parallel plates of the capacitor.
Now, as per the given question, the original capacitance $ c = 3\mu F $ .......... $ \left( 3 \right) $
And, capacitance, when the dielectric is inserted, is $ {c_d} = 15\mu F $ ......... $ \left( 4 \right) $
Dividing equation $ \left( 1 \right) $ and $ \left( 2 \right) $ , we get:
$ \dfrac{c}{{{c_d}}} = \dfrac{{\dfrac{{A{\varepsilon _0}}}{d}}}{{\dfrac{{kA{\varepsilon _0}}}{d}}} $ ............. $ \left( 5 \right) $
$ \dfrac{c}{{{c_d}}} = \dfrac{1}{k} $ ........... $ \left( 6 \right) $
Now, putting the values from the equation $ \left( 3 \right) $ and $ \left( 4 \right) $ in the equation $ \left( 6 \right) $ :
We get, $ \dfrac{{3\mu F}}{{15\mu F}} = \dfrac{1}{k} $
Solving this, we get $ k = 5 $
Now, to calculate the permittivity we will sue the formula $ \varepsilon = {\varepsilon _0}k $
where the value of $ {\varepsilon _0} $ is permittivity constant and it is $ {\varepsilon _0} = 8.85 \times {10^{ - 12}} $
When a dielectric is inserted in between the capacitor plates, the permittivity constant also increased by a factor of $ k $ , so the new permittivity will be
$ \varepsilon = {\varepsilon _0}k $
Putting the value of $ k $ and $ {\varepsilon _0} $ , it becomes
$ = 8.85 \times {10^{ - 12}} \times 5 $
$ \Rightarrow 0.44 \times {10^{ - 10}}{C^2}{N^{ - 1}}{m^{ - 2}} $
Therefore, the answer is an option (C) $ 0.44 \times {10^{ - 10}}{C^2}{N^{ - 1}}{m^{ - 2}} $ .
Note
Remember in such questions it is very obvious to make unit mistakes. So, always take care of units and $ {\varepsilon _0} $ is a constant, its value is fixed in every question and dielectric constant is different for every material, don’t get confused with the word constant.
Complete step by step solution
Let us take the original capacitance to be $ \prime c\prime $ and the new capacitance, when the dielectric is inserted between the plates, be $ {c_d}^{} $ .
Using the formula of capacitance, $ c = \dfrac{{A{\varepsilon _0}}}{d} $ ....... $ \left( 1 \right) $
When a dielectric is inserted between plates, the capacitance of capacitor increases by the factor of $ k $ (dielectric constant) i.e. the new formula is $ {c_d} = \dfrac{{kA{\varepsilon _0}}}{d} $ ............ $ \left( 2 \right) $
where $ A $ is the area of the capacitor is plates and $ d $ is the distance between the two parallel plates of the capacitor.
Now, as per the given question, the original capacitance $ c = 3\mu F $ .......... $ \left( 3 \right) $
And, capacitance, when the dielectric is inserted, is $ {c_d} = 15\mu F $ ......... $ \left( 4 \right) $
Dividing equation $ \left( 1 \right) $ and $ \left( 2 \right) $ , we get:
$ \dfrac{c}{{{c_d}}} = \dfrac{{\dfrac{{A{\varepsilon _0}}}{d}}}{{\dfrac{{kA{\varepsilon _0}}}{d}}} $ ............. $ \left( 5 \right) $
$ \dfrac{c}{{{c_d}}} = \dfrac{1}{k} $ ........... $ \left( 6 \right) $
Now, putting the values from the equation $ \left( 3 \right) $ and $ \left( 4 \right) $ in the equation $ \left( 6 \right) $ :
We get, $ \dfrac{{3\mu F}}{{15\mu F}} = \dfrac{1}{k} $
Solving this, we get $ k = 5 $
Now, to calculate the permittivity we will sue the formula $ \varepsilon = {\varepsilon _0}k $
where the value of $ {\varepsilon _0} $ is permittivity constant and it is $ {\varepsilon _0} = 8.85 \times {10^{ - 12}} $
When a dielectric is inserted in between the capacitor plates, the permittivity constant also increased by a factor of $ k $ , so the new permittivity will be
$ \varepsilon = {\varepsilon _0}k $
Putting the value of $ k $ and $ {\varepsilon _0} $ , it becomes
$ = 8.85 \times {10^{ - 12}} \times 5 $
$ \Rightarrow 0.44 \times {10^{ - 10}}{C^2}{N^{ - 1}}{m^{ - 2}} $
Therefore, the answer is an option (C) $ 0.44 \times {10^{ - 10}}{C^2}{N^{ - 1}}{m^{ - 2}} $ .
Note
Remember in such questions it is very obvious to make unit mistakes. So, always take care of units and $ {\varepsilon _0} $ is a constant, its value is fixed in every question and dielectric constant is different for every material, don’t get confused with the word constant.
Recently Updated Pages
Mark and label the given geoinformation on the outline class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
When people say No pun intended what does that mea class 8 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Name the states which share their boundary with Indias class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give an account of the Northern Plains of India class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Advantages and disadvantages of science
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Bimbisara was the founder of dynasty A Nanda B Haryanka class 6 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
10 examples of evaporation in daily life with explanations
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
How do you graph the function fx 4x class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)